where the sum over atomic plane indexes \(i,j\)  extends over nearest-neighbors only and \(c_{i\sigma }^{\dagger }\)  (\(c_{i\sigma }\) ) creates (annihilates) an electron with spin \(\sigma \)  in the \(i\) -th atomic plane. Each atomic monolayer is assumed to be translationally invariant along the in-plane directions. For simplicity, we assume that all hopping parameters are spin-independent with the same value \(t = -1\)  eV in all regions [1]}, [2]}, [3]}, [4]}, [5]}. With this assumption, the spin dependent block Hamiltonian elements for each layer is given in terms of the spin-dependent onsite energy \(\epsilon _{\sigma }(\textbf {k}_{||})\) , where \(\textbf {k}_{||}\)  refers to the in-plane momenta, and nearest-neighbor hopping parameter \(t\)  as \(H_{ij}^{\sigma \sigma ^{\prime }} = \delta _{\sigma \sigma ^{\prime }}\delta _{ij}\epsilon _{\sigma }(\textbf {k}_{||}) + \delta _{\sigma \sigma ^{\prime }} (\delta _{i, j+1} + \delta _{i, j - 1})t\)  where \(\sigma = \uparrow , \downarrow \)  is the spin index and \(\delta _{ij (\sigma \sigma ^{\prime })}\)  is the Kronecker delta for real (spin) space labels. Assuming the interface is oriented along the \(xy\)  plane, the translation invariance along the in-plane directions allows us to express \(\epsilon _{\sigma }(\textbf {k}_{||}) = \epsilon _{\sigma } + 2t(\cos (k_x) + \cos (k_y))\) .
